This activity is about finding a required period of time when given a repeated percentage change.

You will need a calculator for this activity!

__Recap__

We know that if we invest £3,000 in the bank for 4 years at an interest rate of 3%, to find the total amount after 4 years we would do the calculation:

# £3,000 x 1.03^{4}

*£3,000 being the investment*

*1.03 being the multiplier for an increase of 3%*

*The power 4 being the 4 (periods) years we need to calculate.*

We also know if a car cost £5,000 and it depreciated by 2% every year, to find the value of the car after 5 years we would do the calculation:

# £5,000 x 0.98^{5}

*£5,000 being the initial cost of the car*

*0.98 being the multiplier for an decrease of 2% (100 - 2 = 98)*

*The power 5 being the 5 (periods) years we need to calculate.*

In this activity, we will be given all the information, but we will have to find the number of periods/years/the power.

If you are confused, a look at a typical question will help.

__Example__

Mr Penny invested £4,000 in the bank at an interest rate of 6%.

He wanted to leave it in there until his investment was worth more than £5,000

For how many years would Mr Penny have to leave his money in the bank?

__Answer__

We use the same formula as in the recap:

# 4,000 x 1.06^{n} > 5,000

*£4,000 the investment*

*1.06 being the multiplier for an increase of 6%*

*The power is unknown as we don't know how long it will take, so we write n*

*We do know that we need the total to be > £5,000*

When we have the formula, we use trial and improvement to find n!

Let's try n = 1 year first

# 4,000 x 1.06^{1} = 4,240 (not > 5,000)

Let's try n = 2 years

# 4,000 x 1.06^{2} = 4,494.40 (not > 5,000)

Let's try n = 3 years

# 4,000 x 1.06^{3} = 4,764.06 (not > 5,000)

Getting nearer!

Let's try n = 4 years

# 4,000 x 1.06^{4} = 5,049.91 (this is > 5,000)

Therefore, Mr Penny would need to leave his money in the bank for 4 years to get more than £5,000

*Tip: If it is taking a long time to get anywhere near the amount you need, you can skip some to get there quicker!*

Let's try some of these!